Determining in-band optical signal-to-noise ratio in polarization-multiplexed optical signals using signal correlations

ABSTRACT

A method and apparatus for determining in-band OSNR in optical information signals, e.g. in polarization-multiplexed QPSK and higher-order M-ary QAM signals, are disclosed. A correlation measurement of the signal amplitude or power at two distinct optical frequencies of the signal may be used to determine the in-band optical noise in the signal. A measurement of the signal power may be used to determine the OSNR based on the determined in-band noise.

TECHNICAL FIELD

The present disclosure relates to optical test and measurement, and inparticular to determination of optical signal-to-noise ratio and othersignal quality parameters.

BACKGROUND

The quality of modulated optical signals transmitted in long-distancefiberoptic communication systems is frequently characterized by opticalsignal-to-noise ratio (OSNR), which defines a ratio of the total opticalpower of the digital information signal to optical noise added to thesignal by optical amplifiers. In communication systems with only a fewwidely-spaced wavelength-multiplexed signals, OSNR may be readilydetermined by spectral analysis of a transmitted noisy signal and theoptical noise floor on either side of the signal spectrum. By way ofexample, International Electrotechnical Commission Standards Document“Digital systems—Optical signal-to-noise ration measurements for densewavelength-division multiplexed systems,” IEC 61280-2-9, 2009, describessuch an OSNR measurement.

In modern optical communication systems with dense wavelength-divisionmultiplexing (DWDM), various transmitted optical signals are closelyspaced in optical frequency, so that it becomes difficult to measureoptical noise floor between adjacent signal spectra. This is of aparticular concern for communication systems transmitting opticalsignals at bit rates of 100 Gb/s over 50-GHz wide wavelength channels.In these systems, one needs to measure the optical noise floor withinthe spectral bandwidth of the signal to determine the signal's OSNR.Such measurements are commonly referred to as in-band OSNR measurements.Furthermore, it is frequently required that these in-band OSNRmeasurements are performed while the communication system is in service,i.e. that the noise floor within the signal's bandwidth is determinedwhile the optical information signal is transmitted.

Several methods have been disclosed to measure an in-band OSNR inpresence of transmitted signals. For conventional single-polarizedoptical information signals (e.g. for 10 Gb/s NRZ-OOK signals), apolarization nulling technique has been disclosed. Polarization nullingsubstantially removes the polarized signal from the received noisysignal, thus revealing the floor of an unpolarized optical noise in thespectral bandwidth of the signal. Such a technique has been described in“Optical signal-to-Noise Ratio Measurement in WDM Networks UsingPolarization Extinction” by M. Rasztovits-Wiech et al., EuropeanConference on Optical Communication, 20-24 Sep. 1998, Madrid Spain, pp.549-550.

Modern optical information signals are frequently composed of twomutually orthogonally polarized optical carriers at a same opticalfrequency. The carriers are independently modulated with digitalinformation data. This polarization multiplexing technique is frequentlyused in long-distance communication systems to transmit 50 Gb/s BPSK,100 Gb/s QPSK, or 200 Gb/s 16-QAM signals over 50-GHz wide DWDMchannels. In polarization-multiplexed (PM) signals, in-band OSNR cannotbe determined by means of the above-referenced polarization-nullingtechnique, because the two orthogonally polarized optical carrierscannot be simultaneously removed from the noisy optical signal withoutalso extinguishing the optical noise.

While several methods have been disclosed to measure in-band OSNR inpolarization-multiplexed signals, they generally only work with opticalsignals of a predetermined bit-rate, modulation format, and/or signalwaveform. Consequently, these methods may be suitable for monitoring ofin-band OSNR at certain points in a communication system, e.g. by meansof built-in monitoring equipment, but are difficult to use as a generaltest and measurement procedure. Furthermore, some of these methods arenot suitable for determining in-band OSNR in signals substantiallydistorted by chromatic dispersion (CD) or polarization-mode dispersion(PMD).

By way of example, a method for in-band OSNR measurements, that onlyworks with binary PSK and ASK signals, has been disclosed by X. Liu etal. in “OSNR monitoring method for OOK and DPSK based on optical delayinterferometer,” Photon. Technol. Lett., Vol. 19, p. 1172 (2007), aswell as by W. Chen et al. in “Optical signal-to-noise ratio monitoringusing uncorrelated beat noise,” Photon. Technol. Lett., Vol. 17, p. 2484(2005), and M. Bakaul in “Low-cost PMD-insensitive and dispersiontolerant in-band OSNR monitor based on uncorrelated beat noisemeasurement,” Photon. Technol. Lett., Vol. 20, p. 906 (2008). Thismethod does not work with 100 Gb/s PM-QPSK or 200 Gb/s PM-16-QAMsignals.

Other methods for in-band OSNR monitoring of polarization-multiplexedsignals are based on coherent detection with high-speed receivers andsubsequent digital signal processing. These methods typically operate ata predetermined bit-rate. One of these methods, disclosed by T. Saida etal. in “In-band OSNR monitor with high-speed integrated Stokespolarimeter for polarization division multiplexed signal,” Opt. Express,Vol. 20, p. B165 (2012), determines the in-band OSNR from the spread ofthe four polarization states through which an optical PM QPSK signalcycles rapidly. Clearly, such high-speed polarization analysis requiresprior knowledge of the modulation format and the bit-rate of thetransmitted signal and, furthermore, is very sensitive to signaldistortions caused by chromatic dispersion (CD) and polarization modedispersion (PMD).

For applications in long-distance communication systems, it may beadvantageous to remove CD- and PMD-induced signal distortions prior todetermining OSNR. Compensation of signal distortions introduced by CDand PMD may be accomplished electronically in a high-speed digitalsignal processor 10, which is shown schematically in FIG. 1. In FIG. 1,wavelength-division multiplexed (WDM) signals 11 are coupled to a firstpolarization beamsplitter (PBS) 12 a, and a local oscillator (LO) laser13 is coupled to a second PBS 12 b. The split optical signals are mixedin 90° hybrid mixers 14, converted into electrical signals byphotodetectors 15, and digitized by analog-to-digital converters (ADCs)16. The digital signal processor 10 performs CD compensation 17, PMDcompensation 18, and phase recovery 19. Finally, the OSNR is computed at10 a.

The digital compensation has a disadvantage in that it requires thehigh-speed ADCs 16, which usually have only a relatively small dynamicrange (typically less than 16 dB), thus limiting the OSNR measurementrange. In-band OSNR measurement methods employing error vector magnitude(EVM) analysis of the received signal after electronic compensation ofCD and PMD have been disclosed by D. J. Ives et al. in “Estimating OSNRof equalised QPSK signals,” ECOC 2011 Technical Digest, paper Tu.6.A.6(2011) and by R. Schmogrow et al. in “Error vector magnitude as aperformance measure for advanced modulation formats,” Photon. Technol.Lett., Vol. 24, p. 61 (2012), as well as by H. Y. Choi et al., “OSNRmonitoring technique for PDM-QPSK signals based on asynchronousdelay-tap sampling technique,” OFC 2010 Technical Digest, Paper JThA16.EVM analysis intrinsically requires foreknowledge of the particularmodulation format of the optical signal.

Another method for OSNR monitoring is based on RF spectral analysis oflow-speed intensity variations of polarization-multiplexed signals. Thismethod has been disclosed by H. H. Choi et al. in “A Simple OSNRMonitoring Technique Based on RF Spectrum Analysis for PDM-QPSKSignals,” OECC 2012 Technical Digest (Korea), Paper 6B3-4. However, thismethod is very sensitive to variations in the signal's waveform. Hence,it requires not only foreknowledge of the modulation format and bit-rateof the analyzed optical signal, but also careful calibration with anoiseless signal.

A method for in-band OSNR measurements using conventional spectralanalysis of the optical signal power has been disclosed by D. Gariépy etal. in “Non-intrusive measurement of in-band OSNR of high bit-ratepolarization-multiplexed signals,” Opt. Fiber Technol. vol. 17, p. 518(2011). A disadvantage of this method is that it only works with signalswhose optical spectrum is substantially narrower than the spectral widthof the DWDM channel, e.g. it works with 40 Gb/s PM NRZ-QPSK signalstransmitted through 50-GHz wide DWDM channels, but usually not with 100Gb/s PM RZ-QPSK signals transmitted through 50-GHz wide DWDM channels.

Yet another method for in-band OSNR measurements inpolarization-multiplexed signals has been disclosed by W. Grupp inEuropean Patent EP 2,393,223 “In-band SNR measurement based on spectralcorrelation,” issued Dec. 7, 2011. This method determines in-band OSNRfrom measurements of the cyclic autocorrelation function of the signalamplitude, i.e. by calculating noiseless signal power from correlationsbetween spectral components of the Fourier transform of the cyclicautocorrelation function. The cyclic autocorrelation function of thesignal's amplitude may be measured, for example, by means of twoparallel coherent receivers employing a common pulsed local oscillatorlaser, as shown schematically in FIG. 2. A pulsed laser 20 is triggeredby a clock recovery circuit 21, which receives light via a tap 21 acoupled to a 3 dB splitter 22. Electrical signals of the photodetectors15 are filtered by low-pass filters 23. A variable optical delay 24 isemployed to sample the optical signal 11 twice within each symbol periodat various times and with various differential delays between the twosamples taken within the same symbol period. A processor 25 is used tocalculate the OSNR. This method also requires foreknowledge of themodulation format and bit-rate of the optical signal, as well as carefulcalibration of the apparatus with a noiseless signal. In addition, themethod is very sensitive to signal distortions introduced by CD and/orPMD.

SUMMARY

According to one aspect of the disclosure, there is provided a methodand apparatus for in-service determination of in-band OSNR inpolarization-multiplexed, as well as single-polarized signals of unknownbitrate and modulation format. The method may allow accuratedetermination of the in-band OSNR not only in binary ASK and PSK encodedsignals, but also in conventional QPSK and M-ary QAM signals. It isrelatively insensitive to large CD- or PMD-induced signal distortions,and does not require prior calibration with a similar noiseless opticalsignal.

These highly desirable features may be accomplished by first measuringan optical power spectrum of a noisy signal, e.g. by means of aconventional optical spectrum analyzer, and by subsequently measuringcorrelations between predetermined pairs of spaced apart time-varyingfrequency components in the optical amplitude or power/intensityspectrum of the signal by means of two optically narrow-band amplitudeor power detectors. The power detectors may include narrow-band coherentreceivers with phase and polarization diversity. The coherent receiversmay include continuous wave (cw), not pulsed laser(s) as localoscillator(s) (LO).

In accordance with an embodiment of the disclosure, there is provided amethod for determining an optical signal-to-noise ratio of a modulatedoptical signal propagating in a transmission link, the modulated opticalsignal comprising a plurality of wavelength channels, the methodcomprising:

(a) measuring an optical power spectrum of the modulated optical signal,the optical power spectrum comprising at least one of the plurality ofwavelength channels;

(b) measuring a time-varying parameter comprising at least one of:time-varying optical signal amplitudes and phases in two mutuallyorthogonal polarization states; and time-varying optical signal powerlevels in two mutually orthogonal polarization states; wherein thetime-varying parameter is measured simultaneously at first and secondpredetermined optical frequencies in a selected one of the plurality ofwavelength channels, wherein the first and second predetermined opticalfrequencies are separated by a non-zero frequency interval;

(c) determining a correlation between the time-varying parametersmeasured in step (b) at the first and second optical frequencies bycalculating a correlation coefficient between the time-varyingparameters at the first and second optical frequencies; and

(d) determining the optical signal-to-noise ratio from the optical powerspectrum measured in step (a) and the correlation coefficient calculatedin step (c).

In accordance with an aspect of the disclosure, there is provided amethod for determining a group velocity dispersion accumulated due tochromatic dispersion of a modulated optical signal comprising aplurality of wavelength channels, the method comprising:

(a) measuring time-varying amplitudes and phases of the modulatedoptical signal in two mutually orthogonal polarization statessimultaneously at first and second predetermined optical frequenciesseparated by a non-zero frequency interval, in at least one of theplurality of wavelength channels;

(b) introducing a differential time and phase delay between signalsrepresenting the time-varying optical signal amplitudes and phases atthe first and second optical frequencies;

(c) determining a correlation between the time-varying optical signalamplitudes and phases at the predetermined optical frequencies bycalculating a correlation coefficient between the time-varyingamplitudes and phases of the modulated optical signal; and

(d) varying the differential time and phase delay of step (b);

(e) repeating steps (c) and (d) until the correlation coefficientreaches a maximum; and

(f) calculating the group velocity dispersion from the differential timeand phase delay introduced in step (b) and varied in step (d), and thefrequency interval of step (a).

In accordance with an aspect of the disclosure, there is provided amethod for determining a differential group delay accumulated due topolarization mode dispersion of a modulated optical signal comprising aplurality of wavelength channels, the method comprising:

(a) measuring time-varying optical signal power of the modulated opticalsignal in two mutually orthogonal polarization states simultaneously atfirst and second predetermined optical frequencies separated by anon-zero frequency interval, in at least one of the plurality ofwavelength channels;

(b) introducing a differential group delay between signals representingthe time-varying optical signal powers at the first and second opticalfrequencies;

(c) determining a correlation between the time-varying optical signalpowers at the first and second optical frequencies by calculating acorrelation coefficient between the time-varying optical signal powers;and

(d) varying the differential group delay of step (b);

(e) repeating steps (c) and (d) until the correlation coefficientreaches a maximum;

(f) using the last value of the differential group delay varied in step(d) to obtain the differential group delay accumulated due topolarization mode dispersion of the modulated optical signal.

In accordance with an aspect of the disclosure, there is provided anapparatus for determining an optical signal-to-noise ratio of amodulated optical signal comprising a plurality of wavelength channels,the apparatus comprising:

a spectrum analyzer for measuring an optical power spectrum of themodulated optical signal, the optical power spectrum comprising at leastone of the plurality of wavelength channels;

a frequency selective splitter for selecting first and second portionsof the modulated optical signal at first and second predeterminedoptical frequencies, respectively, in a selected one of the plurality ofwavelength channels, wherein the first and second predetermined opticalfrequencies are separated by a non-zero frequency interval;

a measuring unit for measuring a time-varying parameter comprising atleast one of: time-varying optical amplitudes and phases; andtime-varying optical power levels of the first and second portions ofthe modulated optical signal;

a signal processor for determining a correlation between thetime-varying parameters of the first and second portions of themodulated optical signal, and for calculating the opticalsignal-to-noise-ratio from the correlation of the time-varyingparameters and the power spectrum of the modulated optical signal.

In embodiments where the time-varying parameter includes thetime-varying optical signal amplitudes and phases, the amplitude andphase detector may include a coherent receiver, and the frequencyselector may include a tunable local oscillator light source, e.g. anarrowband cw laser source.

BRIEF DESCRIPTION OF THE DRAWINGS

Exemplary embodiments will now be described in conjunction with thedrawings, in which:

FIG. 1 is a schematic illustration of a prior-art apparatus formeasuring in-band OSNR using a high-speed coherent receiver and adigital signal processor;

FIG. 2 is a schematic illustration of a prior-art apparatus formeasuring in-band OSNR by means of high-speed optical sampling of thesignal amplitude with two parallel coherent receivers employing a pulsedlocal oscillator laser at a single optical frequency;

FIG. 3 illustrates a plot of an optical power spectrum of a noisy 100Gb/s polarization-multiplexed RZ-QPSK signal, showing spectralcomponents used for measuring the correlation coefficients at two spacedapart optical frequencies;

FIG. 4 illustrates a plot of a normalized amplitude spectral correlationdensity function (SCDF) evaluated at two spaced apart opticalfrequencies and of the OSNR estimated form the SCDF versus “true”in-band OSNR in a 100 Gb/s polarization-multiplexed RZ-QPSK signal;

FIG. 5 illustrates a plot of an optical power spectrum of a noisy 100Gb/s polarization-multiplexed RZ-QPSK signal, showing an example of thespectral components used for correlation measurements, in which thecenter frequency f is offset from the carrier frequency f_(c) of thesignal;

FIG. 6 illustrates a plot of a normalized amplitude SCDF and estimatedOSNR versus frequency offset f−f_(c) for a 100 Gb/spolarization-multiplexed RZ-QPSK signal;

FIG. 7 illustrates a plot of optical power variations in two spectralcomponents of a substantially noiseless 100 Gb/s PM-QPSK signal (>40 dBOSNR), measured at two spaced apart optical frequencies, with an opticalbandwidth of 200 MHz;

FIG. 8 illustrates a plot of optical power variations in two spectralcomponents of a noisy 100 Gb/s PM-QPSK signal having an OSNR of only 12dB OSNR, measured at two spaced apart optical frequencies, with anoptical bandwidth of 200 MHz;

FIG. 9 illustrates a plot of optical power variations of two spectralcomponents of a noiseless 100 Gb/s PM-QPSK signal distorted by chromaticdispersion with 50,000 ps/nm GVD;

FIG. 10 illustrates a plot of the normalized power SCDF at two spacedapart optical frequencies centered around f_(c), and the estimated OSNRcalculated for a 100 Gb/s PM RZ-QPSK signal;

FIG. 11 illustrates a plot of the normalized power SCDF and theestimated OSNR versus frequency offset f−f_(c) calculated for a 100 Gb/sPM RZ-QPSK signal;

FIG. 12 is a schematic illustration of an apparatus for measuringamplitude correlations in the spectrum of a modulated signal, using twoparallel coherent receiver channels with phase and polarizationdiversity and subsequent digital signal processing;

FIG. 13 illustrates a plot of OSNR estimated from amplitude SCDF in acoherently received 100 Gb/s PM RZ-QPSK signal versus reference OSNR,calculated for a receiver with 5^(th)-order Butterworth electriclow-pass filter having four different bandwidths;

FIG. 14 illustrates a plot of a normalized amplitude and power SCDFs ina 100 Gb/s PM RZ-QPSK signal versus frequency separation of localoscillator (LO) lasers, displayed versus frequency deviation. The solidcurves assume an electrical receiver with 5^(th)-order Butterworthlow-pass filter and the dashed curve a receiver with 5^(th)-order Bessellow-pass filter, both having a 3-dB bandwidth of 40 MHz;

FIG. 15 is a schematic illustration of an apparatus for measuring theoptical power spectrum of the noisy signal, using a coherent receiverwith continuously tunable LO laser;

FIG. 16 is a schematic illustration of an apparatus for measuringintensity correlations in the spectrum of a modulated signal, using twoparallel coherent receivers with phase and polarization diversity andfast digital signal processing;

FIG. 17 is a schematic illustration of another apparatus embodiment formeasuring amplitude or intensity correlations in the optical spectrum ofa modulated signal, using two parallel heterodyne receivers withpolarization diversity and digital down-conversion with phase diversityin a digital signal processor;

FIG. 18 is a schematic illustration of another apparatus embodiment formeasuring amplitude or intensity correlations in the optical spectrum ofa modulated signal, using two parallel coherent receivers and a singlelaser in combination with an optical frequency shifter to generate thetwo optical local oscillator signals spaced apart in frequency;

FIGS. 19A-19C are schematic illustrations of three exemplary embodimentsof optical frequency shifters for generating one or two frequencyshifted optical signals from a single-frequency optical input signal;

FIG. 20 is a schematic illustration of an apparatus embodiment formeasuring intensity correlations in the optical spectrum of a modulatedsignal using two narrowband optical band-pass filters and non-coherentphoto-receivers;

FIG. 21 illustrates a flow chart of a method for determining an opticalsignal-to-noise ratio of a modulated optical signal;

FIG. 22 illustrates a flow chart of a method for determining a groupvelocity dispersion accumulated due to chromatic dispersion of amodulated optical signal; and

FIG. 23 illustrates a flow chart of a method for determining adifferential group delay accumulated due to polarization mode dispersionof a modulated optical signal.

DETAILED DESCRIPTION

While the present teachings are described in conjunction with variousembodiments and examples, it is not intended that the present teachingsbe limited to such embodiments. On the contrary, the present teachingsencompass various alternatives and equivalents, as will be appreciatedby those of skill in the art.

Amplitude and phase of digitally modulated optical signals, such asQPSK- and 16-QAM-modulated signals, may vary pseudo-randomly with time.These pseudo-random amplitude and phase variations are difficult todistinguish from random amplitude and phase variations of optical ASEnoise generated by optical amplifiers, in particular if the waveform ofthe modulated signal is substantially distorted by large amounts ofchromatic dispersion or polarization-mode dispersion in the fiber link.However, an autocorrelation function of digitally modulated signals isperiodic in time, because the transmitted symbols are assignedpredetermined and substantially equal time intervals, whereas theautocorrelation function of random ASE noise does not exhibit suchperiodicity. The periodicity of the autocorrelation function ofdigitally modulated signals is manifested in the signal's opticalfrequency spectrum, which exhibits strong correlations betweentime-varying amplitudes, and also between time-varying intensities andoptical power levels, of certain pairs of spaced apart spectralcomponents, whereas such correlations do not exist in the opticalspectrum of random ASE noise. It is possible, therefore, to determine arelative amount of random ASE noise in a transmitted modulated signal bymeasuring correlations between the aforementioned spaced apart spectralcomponents, and by subsequently comparing the measured correlations tocorresponding correlations in a noiseless signal spectrum. Once therelative amount of ASE noise in a transmitted signals is determined, thein-band OSNR of the noisy signal may be calculated.

Correlations between various spectral components of a digitallymodulated signal may be described by a spectral correlation densityfunction (SCDF), S_(x) ^(α)(f), which is defined as the Fouriertransformation of the cyclic autocorrelation function, R_(x) ^(α)(τ), ofthe optical signal amplitude x(t), i.e.

S_(x)^(α)(f) ≡ ∫_(−∞)^(∞)R_(x)^(α)(τ) ⋅ exp (−j 2 π f τ) d τ

wherein R_(x) ^(α)(τ) is the cyclic auto-correlation function given by

R _(x) ^(α)(τ)≡

x(t+τ/2)·x*(t−τ/2)·exp(−j2παt)

,

and x(t) is a time-varying complex two-dimensional Jones vector, whichdescribes amplitude and phase variations of the two polarizationcomponents of the modulated signal as a function of time t. The brackets< > denote time averaging over a time period that is substantiallylonger than the symbol period T_(symbol) of the digital modulation. Moregenerally, the averaging period, and accordingly the measurements ofspectral components amplitudes, phases, and/or optical powerlevels/intensities, should be sufficiently long to ensure apre-determined level of fidelity of computed correlations.

Equivalently, the SCDF may be expressed as a correlation function of thetime-varying amplitudes of the spectral components of the modulatedsignal, i.e. as

S _(x) ^(α)(f)=

X _(T)(t,f+α/2)·X _(T)*(t,f−α/2)

wherein

X_(T)(t, v) = ∫_(t − T/2)^(t + T/2)x(u) ⋅ exp (−j 2 π vu) du

and T is an integration time with T>>T_(symbol). The brackets in theabove expression denote averaging over a time period substantiallylonger than the integration time T. For methods disclosed herein, it maybe advantageous to define a normalized SCDF of the spectralcorrelations, i.e.

${{{\hat{S}}_{x}^{\alpha}(f)} = \frac{\langle{{X_{T}\left( {t,{f + {\alpha/2}}} \right)} \cdot {X_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}}\rangle}{\sqrt{\langle{{X_{T}\left( {t,{f + {\alpha/2}}} \right)}}^{2}\rangle}\sqrt{\langle{{X_{T}\left( {t,{f - {\alpha/2}}} \right)}}^{2}\rangle}}},$

which is similar to the un-balanced correlation coefficient used instatistical analysis, and which has the property −1≤Ŝ_(x) ^(α)(f)≤1 forall values of f and α.

It is known that noiseless and otherwise undistorted optical signalsencoded with binary amplitude-shift keying (ASK), binary phase-shiftkeying (BPSK), ordinary quaternary phase-shift keying (QPSK), and16-quadrature-amplitude modulation (16-QAM) exhibit Ŝ_(x) ^(α)(f)=1 whenα=α₀=1/T_(symbol) and for all frequencies f within the range−α/2<f<−α/2. In addition, it is known that Ŝ_(x) ^(α)(f)=1 whenα=2/T_(symbol). However, it is important to note that for opticalsignals encoded with staggered QPSK modulation, also referred to as“offset QPSK”, Ŝ_(x) ^(α)(f)≈0 when α=1/T_(symbol). For staggered QPSKŜ_(x) ^(α)(f)=1 only when α=²/T_(symbol).

In contrast to modulated optical signals, optical ASE noise is a randomGaussian process and, therefore, does not exhibit any significantcorrelation between its spectral components. Therefore, when random ASEnoise is added to a modulated optical signal, the normalized SCDF isalways smaller than unity, i.e. S_(x) ^(α)(f)<1, as described below inmore detail. Let n(t) denote the Jones vector of the phase and amplitudeof random ASE noise added to the transmitted signal, then the Jonesvector {tilde over (X)}_(T)(t,v) of the spectral component of the noisysignal at frequency v is the sum of the noiseless Jones vectorX_(T)(t,v), defined above, and the corresponding Jones vector of the ASEnoise N_(T)(t,v), i.e.

${{\overset{\sim}{X}}_{T}\left( {t,v} \right)} = {{\int_{t - {T/2}}^{t + {T/2}}{{\left( {{x(u)} + {n(u)}} \right) \cdot {\exp \left( {{- j}\; 2\; \pi \; {vu}} \right)}}{du}}} \equiv {{X_{T}\left( {t,v} \right)} + {{N_{T}\left( {t,v} \right)}.}}}$

Consequently, the SCDF of a noisy modulated signal can be expressed as

$\begin{matrix}{S_{x}^{\alpha {(f)}} = {\langle{{{\overset{\sim}{X}}_{T}\left( {t,{f + {\alpha/2}}} \right)} \cdot {{\overset{\sim}{X}}_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}}\rangle}} \\{= {{\langle{{X_{T}\left( {t,{f + {\alpha/2}}} \right)} \cdot {X_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}}\rangle} +}} \\{{\langle{{N_{T}\left( {t,{f + {\alpha/2}}} \right)} \cdot {N_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}}\rangle}}\end{matrix}$

wherein the second term on the right side of the equation vanishes when|α|>0. It should be noted that

X _(T)(t,f+α/2)·N _(T)*(t,f−α/2)

=

X _(T)(t,f+α/2)·N _(T)*(t,f−α/2)

≡0,

because the amplitudes of random noise and modulated signal areuncorrelated. Thus, the normalized SCDF for α>0, e.g. forα=1/T_(symbol), is given by

${{\hat{S}}_{x}^{\alpha}(f)} = \frac{\langle{{X_{T}\left( {t,{f + {\alpha/2}}} \right)} \cdot {X_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}}\rangle}{\begin{matrix}\sqrt{\langle{{{X_{T}\left( {t,{f + {\alpha/2}}} \right)}}^{2} + {{N_{T}\left( {t,{f + {\alpha/2}}} \right)}}^{2}}\rangle} \\\sqrt{\langle{{{X_{T}\left( {t,{f - {\alpha/2}}} \right)}}^{2} + {{N_{T}\left( {t,{f - {\alpha/2}}} \right)}}^{2}}\rangle}\end{matrix}}$

Assuming that the power spectrum of the modulated signal is symmetricabout its carrier frequency f_(c), which is generally the case for theaforementioned modulation formats, so that

|X _(T)(t,f _(c)+α/2)|²

=

|X _(T)(t,f _(c)−α/2)|²

≡

P _(S)(f _(c)±α/2)

,

and further assuming that the power spectrum of the random ASE also issubstantially symmetric about f_(c), which is frequently the case, sothat

|N _(T)(t,f+α/2)|²

=

|N _(T)(t,f−α/2)|²

≡

P _(N)(f _(c)±α/2)

,

then the normalized SCDF at difference frequency α₀=1/T_(symbol) can beexpressed as

${{{\hat{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)} = \frac{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle}{{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} + {\langle{P_{N}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle}}},$

which—after rearrangement—yields the signal-to-noise ratio, SNR, of thespectral components at the two optical frequencies f_(c)−α₀/2 andf_(c)+α₀/2 as

${{S\; N\; {R\left( {f_{c},\alpha_{0}} \right)}} = {\frac{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle}{\langle{P_{N}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} = \frac{{\hat{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)}{1 - {{\hat{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)}}}},$

and, similarly, the ratio of the total signal and noise power to thenoise power at f_(c)±α₀/2 as

$\frac{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + {P_{N}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}}\rangle}{\langle{P_{N}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} = {\frac{1}{1 - {{\hat{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)}}.}$

It is known to those skilled in the art that the OSNR of a transmittedsignal is defined as the ratio of the total signal power over the totalnoise power in an optical bandwidth B_(noise) (usually equal to 0.1 nm;c.f. IEC 61280-2-9 “Digital systems—Optical signal-to-noise ratiomeasurements for dense wavelength-division multiplexed systems”) as

${{{OSNR} \equiv \frac{\sum\limits_{i}{{\langle{P_{S}\left( f_{i} \right)}\rangle}B_{meas}}}{{\langle P_{N}\rangle}B_{noise}}} = {\frac{\sum\limits_{i}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle P_{N}\rangle}B_{noise}} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}},$

wherein the summation extends over all frequency components f_(i) withinthe bandwidth of the signal, and B_(meas) denotes the measurementbandwidth of each power measurement

P_(S)(f_(i))

and

P_(N)(f_(i))

. It should be noted that the above definition of OSNR assumes that thespectrum of the random ASE noise is substantially flat within thebandwidth of the modulated signal, so that the average noise powerdensity is identical at all frequencies, i.e.

P_(N)(f_(i))

=

P_(N)(f_(j))

=

P_(N)

for all frequencies f_(i)≠f_(i) within the bandwidth of the modulatedoptical signal. The in-band OSNR of a noisy signal may thus bedetermined from a measurement of the power spectrum of the transmittednoisy signal, i.e. of

P_(S)(f)

+

P_(N)(f)

, and an additional measurement of the average noise power,

P_(N)(f)

, within the bandwidth of the modulated signal.

Whereas conventional single-polarized signals

P_(N)

may be directly measured by blocking the polarized signal with aproperly oriented polarization filter, such measurements cannot beperformed with polarization-multiplexed signals. In the presentdisclosure,

P_(N)

may be obtained from the spectral correlation of the complex Jonesvectors of the signal amplitudes at optical frequencies f_(c)−α₀/2 andf_(c)+α₀/2, as described above, and expressed as a fraction of the noisysignal power at these frequencies, i.e. as

P _(N)

=

P _(S)(f _(c)±α₀/2)+P _(N)

└1−Ŝ _(x) ^(α) ⁰ (f _(c))┘.

It should be noted that this procedure can be applied topolarization-multiplexed signals, as well as to single-polarizedsignals. Substituting the above relation into the equation for OSNRimmediately yields the desired in-band OSNR if the optical signal,

$\begin{matrix}{{OSNR} = {{\frac{\sum\limits_{i}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}} \cdot \frac{1}{1 - {S_{x}^{\alpha_{0}}\left( f_{c} \right)}}} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}} \\{{= \frac{\sum\limits_{i}{\begin{Bmatrix}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle} -} \\{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}\left\lbrack {1 - {{\hat{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)}} \right\rbrack}\end{Bmatrix}B_{meas}}}{{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}\left\lbrack {1 - {{\hat{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)}} \right\rbrack}B_{noise}}},}\end{matrix}$

wherein all quantities are known from the two measurements describedabove.

Therefore, the in-band OSNR of a transmitted noisy signal may bedetermined from a measurement of the spectral correlation of the opticalamplitudes at frequencies f_(c)±α₀/2 and a conventional spectralanalysis of the combined signal and noise power, as shown in the exampleof FIG. 3 for a 100 Gb/s PM-QPSK noisy signal 30 compared with anoiseless signal 31. Spectral components 32 and 33 may be used formeasuring the correlation coefficients at optical frequencies f_(c)−α₀/2and f_(c)+α₀/2. The spectral components 32 and 33 are preferablyselected so that differences between each of optical frequencies of thespectral components 32 and 33 and the carrier frequency f_(c) of themodulated optical signal in the selected wavelength channel aresubstantially of equal magnitude α₀/2, so that the frequency interval issubstantially equal to the symbol repetition frequency, or an integermultiple of the symbol repetition frequency. More generally, any twopredetermined optical frequencies in a selected one of the plurality ofwavelength channels may be used, provided that the optical frequenciesare separated by a non-zero frequency interval.

The above described method does not require foreknowledge of thetime-varying waveform or the particular modulation format of thetransmitted optical signal. Therefore, determination of the in-band OSNRdoes not require any calibration with a noiseless signal e.g. thenoiseless signal 31. The only foreknowledge required for this method isthat the noiseless signal exhibits a spectral amplitude correlation atfrequencies f_(c)±α₀/2 with Ŝ_(x) ^(α) ⁰ (f_(c))=1. Whereas it may beadvantageous to have foreknowledge of the symbol period T_(symbol) ofthe modulated signal, in order to determine the frequencies f_(c)±α₀/2,this information is not required when Ŝ_(x) ^(α)(f_(c)) is measured at amultitude of frequency pairs f_(c)±α/2, with α ideally ranging from 0 tothe largest possible value, and when

P_(N)

is determined from the maximal value of Ŝ_(x) ^(α)(f_(c)) observed inthis multitude of measurements. Advantageously, the frequency rangewhere Ŝ_(x) ^(α(f) _(c)) is expected to be maximal may be determinedfrom a simple analysis of the signal's power spectrum. Preferably, thefrequency interval between the measured spectral components, e.g. thespectral components 32 and 33 of FIG. 3, is substantially equal to thesymbol repetition frequency of the modulated optical signal in theselected wavelength channel, or an integer multiple thereof. It isfurther preferable that differences between each of the first and secondoptical frequencies of the measured spectral components, and the carrierfrequency f_(c) of the modulated optical signal in the selectedwavelength channel are substantially of equal magnitude.

The in-band OSNR measurement method of the present disclosure may beapplied to transmitted signals that are encoded with chirp-free ASK,BPSK, ordinary QPSK, and other higher-order M-ary QAM formats withoutrequiring detailed knowledge of the particular modulation format encodedin the analyzed noisy signal. Referring to FIG. 4, a numericalsimulation of the in-band OSNR determined from the above equation isshown for a 100 Gb/s polarization-multiplexed QPSK signal. It can beseen from FIG. 4 that an estimated in-band OSNR 40 is substantiallyequal to the reference OSNR over a range from at least 0 dB to 30 dB. Anormalized SCDF 41 is plotted for a reference.

According to an aspect of the disclosure, the in-band OSNR of atransmitted noisy signal may be determined from a measurement of thespectral correlation of the optical amplitudes at any combination offrequencies f±α/2 for which Ŝ_(x) ^(α) ⁰ (f)=1. It is known thatdigitally modulated signals encoded with ASK, BPSK, QPSK, or 16-QAM, forexample, exhibit Ŝ_(x) ^(α) ⁰ (f)=1 as long as the two opticalfrequencies f−α₀/2 and f+α₀/2 are within the optical bandwidth of themodulated signal. This is illustrated in FIG. 5 for the example of a 100Gb/s PM RZ-QPSK noisy signal 50, compared with a noiseless signal 51. InFIG. 5, optical frequencies 52 and 53, at which the time-varying signalmeasurements are performed, are offset from the carrier frequency f_(c).However, it should be noted that the bandwidth of the transmittedoptical signal may be limited by the spectral width of the DWDM channel.For example, when 50 Gb/s PM-BPSK, 100 Gb/s PM-QPSK, or 200 Gb/sPM-16-QAM signals (all having T_(symbol)=40 ps) are transmitted througha 50-GHz wide DWDM channel, the useful values of f are restricted to therange f_(c)−α₀/2<f<f_(c)+α₀/2.

In the case of f≠f_(c) as shown in FIG. 5, one generally has

P_(S)(f+α₀/2)

≠

P_(S)(f−α₀/2)

, so that the process of determining the OSNR becomes significantly morecomplicated. Denoting

P_(S)(f+α₀/2)

=C

P_(S)(f−α₀/2)

≡C

P_(S)

, with C>0 being a real number, the normalized SCDF is given by

${{{\hat{S}}_{x}^{\alpha_{0}}(f)} = \frac{\sqrt{C}{\langle P_{S}\rangle}}{\sqrt{\langle{{CP}_{S} + P_{N}}\rangle}\sqrt{\langle{P_{S} + P_{N}}\rangle}}},$

which may be solved analytically or numerically for

P_(N)

/(

P_(S)

+

P_(N)

).

However, at large OSNR values, one has

P_(S)

>>

P_(N)

, so that the normalized SCDF may be approximated as

${{{\hat{S}}_{x}^{\alpha_{0}}(f)} \approx \frac{{\langle P_{S}\rangle}^{2}}{{\langle P_{S}\rangle}^{2} + {\frac{C + 1}{2C}{\langle P_{S}\rangle}{\langle P_{N}\rangle}} + \frac{{\langle P_{N}\rangle}^{2}}{2C}}},$

from which the average noise power

P_(N)

at the two frequencies f±α₀/2 can be readily calculated as

$\begin{matrix}{\frac{\langle P_{N}\rangle}{\langle{P_{S}\left( {f - {\alpha_{0}/2}} \right)}\rangle} = {{- \frac{C + 1}{2}} + \sqrt{\left( \frac{C + 1}{2} \right)^{2} + {2C\frac{1 - {{\hat{S}}_{x}^{\alpha_{0}}(f)}}{{\hat{S}}_{x}^{\alpha_{0}}(f)}}}}} \\{{\approx {\frac{2C}{C + 1} \cdot \frac{1 - {{\hat{S}}_{x}^{\alpha_{0}}(f)}}{{\hat{S}}_{x}^{\alpha_{0}}(f)}}},}\end{matrix}$

or more conveniently as

${\frac{\langle P_{N}\rangle}{\langle{{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}}\rangle} \approx \frac{2{C\left\lbrack {1 - {{\hat{S}}_{x}^{\alpha_{0}}(f)}} \right\rbrack}}{{2C} + {\left( {1 - C} \right){{\hat{S}}_{x}^{\alpha_{0}}(f)}}}},$

so that one obtains the in-band OSNR as

$\begin{matrix}{{OSNR} \approx {{\frac{\sum\limits_{i}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle{{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}} \cdot \frac{1 + {\frac{1 - C}{2C}{{\hat{S}}_{x}^{\alpha_{0}}(f)}}}{1 - {{\hat{S}}_{x}^{\alpha_{0}}(f)}}} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}} \\{= \frac{\sum\limits_{i}{\begin{Bmatrix}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle} -} \\{{\langle{{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}\frac{2{C\left\lbrack {1 - {{\hat{S}}_{x}^{\alpha_{0}}(f)}} \right\rbrack}}{{2C} + {\left( {1 - C} \right){{\hat{S}}_{x}^{\alpha_{0}}(f)}}}}\end{Bmatrix}B_{meas}}}{{\langle{{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}\frac{2{C\left\lbrack {1 - {{\hat{S}}_{x}^{\alpha_{0}}(f)}} \right\rbrack}}{{2C} + {\left( {1 - C} \right){{\hat{S}}_{x}^{\alpha_{0}}(f)}}}B_{noise}}}\end{matrix}$

Therefore, in-band OSNR of a transmitted noisy signal can be determinedfrom a measurement of the spectral correlation of the optical amplitudesat arbitrary frequencies f±α₀/2 and an additional measurement of theoptical power spectrum of the noisy signal. Again, this procedure doesnot require foreknowledge of the particular modulation format, bit-rate,or time-varying waveform of the transmitted signal. If the symbol period1/α₀ of the signal is unknown, one may simply choose a suitable fixedvalue for f−α₀/2, and then measure Ŝ_(x) ^(α)(f) at a multitude offrequency pairs {f−α₀/2,f+α/2}, with α ranging from 0 to the largestexpected value.

P_(N)

may then be determined from a maximal value of Ŝ_(x) ^(α)(f_(c))observed in these measurements.

However, accurate determination of the in-band OSNR may requireforeknowledge of the ratio C of the two noiseless signal powers atf±α₀/2, which may be determined from a measurement of the optical powerspectrum of the noiseless signal (e.g. directly after the transmitter).If such measurement is not available, one may use the first-orderapproximation

${{C \approx C_{1}} = \frac{\langle{{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}{\langle{{P_{S}\left( {f + {\alpha_{0}/2}} \right)} + P_{N}}\rangle}},$

to estimate the in-band OSNR in the signal,

${OSNR} \approx {{\frac{\sum\limits_{i}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle{{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}} \cdot \frac{1 + {\frac{1 - C_{1}}{2C_{1}}{{\hat{S}}_{x}^{\alpha_{0}}(f)}}}{1 - {{\hat{S}}_{x}^{\alpha_{0}}(f)}}} - {\sum\limits_{i}{\frac{B_{meas}}{B_{noise}}.}}}$

Referring to FIG. 6, an example is shown of OSNR curves 60A and 60Bestimated from a numerically simulated noisy 100 Gb/s PM-QPSK signalhaving an OSNR of 15 dB. The graph also displays a normalized amplitudeSCDF 61 as a function of the frequency offset f−f_(c). OSNR 60A has beencalculated from the above equation using the first-order approximationC≈C₁ (bold the SCDF 61 and the OSNR 60A are shown by solid curves). Itcan be seen that this approximation slightly overestimates the OSNR atoffset frequencies beyond 8 GHz, where the signal power

P_(S)(f+α₀/2)

is small and comparable to the noise power

P_(N)

, as can be seen in FIG. 5. For comparison, the dashed curve 60B in FIG.6 displays the in-band OSNR calculated from the exact formula for C=1.It is evident from this curve that this formula may lead to asubstantial overestimation of the OSNR, e.g. by up to and more than 2dB.

If the first-order approximation of C is not deemed to be accurateenough to determine the in-band OSNR, a second-order approximation maybe employed, which may be obtained by subtracting the first-orderaverage noise power

${\langle P_{N\; 1}\rangle} \approx {{\langle{{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}\frac{2{C_{1}\left\lbrack {1 - {{\hat{S}}_{x}^{\alpha_{0}}(f)}} \right\rbrack}}{{2C_{1}} + {\left( {1 - C_{1}} \right){{\hat{S}}_{x}^{\alpha_{0}}(f)}}}}$

from the two noisy signal powers measured at frequencies f±α₀/2 and byrecalculating C as

${{C \approx C_{2}} = \frac{{\langle{{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}}\rangle} - {\langle P_{N\; 1}\rangle}}{{\langle{{P_{S}\left( {f + {\alpha_{0}/2}} \right)} + P_{N}}\rangle} - {\langle P_{N\; 1}\rangle}}},$

which may then be used for an improved second-order approximation of thein-band OSNR,

${OSNR} \approx {{\frac{\sum\limits_{i}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle{{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}} \cdot \frac{1 + {\frac{1 - C_{2}}{2C_{2}}{{\hat{S}}_{x}^{\alpha_{0}}(f)}}}{1 - {{\hat{S}}_{x}^{\alpha_{0}}(f)}}} - {\sum\limits_{i}{\frac{B_{meas}}{B_{noise}}.}}}$

The above described procedure may be iterated multiple times until thedesired accuracy of the in-band OSNR is obtained.

Amplitudes of the spectral components of a transmitted signal may dependquite sensitively on signal distortions caused by chromatic dispersion(CD) or polarization-mode dispersion (PMD), which the modulated signalmay have experienced in the transmission link prior to being analyzed atthe OSNR monitoring point. Group velocity dispersion (GVD) from CD, forexample, may introduce a differential phase shift between the spectralamplitudes of the Jones vectors X_(T)(t,f+α/2) and X_(T)(t,f−α/2),whereas PMD-induced differential group delays (DGDs) may introduce adifferential polarization transformation between the two Jones vectors.Consequently, the correlation between the two spectral components maybecome severely distorted, so that Ŝ_(x) ^(α) ⁰ (f)<1 even for noiselesssignals. Therefore, uncompensated GVD and/or DGD in the noisy signal maylead to a substantial underestimation of the in-band OSNR when using thespectral correlation method disclosed above.

Fortunately, the differential phase shifts caused CD and thedifferential polarization transformation caused by PMD may becompensated for by artificially introducing differential phase shiftsand/or differential polarization transformations in the measured Jonesvectors X_(T)(t,f+α₀/2) and X_(T)(t,f−α₀/2), and by varying these phaseshifts and/or polarization transformations until, Ŝ_(x) ^(α) ⁰ (f) ismaximal. To that end, the step of determining the correlation mayinclude (i) removing differential phase and time delays introduced bychromatic dispersion in the transmission link between the time-varyingparameters at the optical frequencies, at which the measurement isperformed; and (ii) removing a differential group delay introduced bypolarization mode dispersion in the transmission link between thetime-varying parameters at the first and second optical frequencies.Even if the GVD- and DGD-induced distortions in the signal amplitudesare not perfectly compensated, such procedure may substantially reduceerrors of determining the in-band OSNR from the measured spectralcorrelation.

In another aspect of the present disclosure, the end-to-end GVD and DGDin the transmission link may be determined from an adaptive compensationof the GVD and DGD in the measured signal amplitudes using the abovedescribed algorithm for maximizing Ŝ_(x) ^(α) ⁰ (f).

The in-band OSNR may also be determined from the spectral correlationsof the optical signal intensities (or signal powers) at two differentfrequencies, i.e. from the correlations between the spectral powercomponents {tilde over (P)}(t,v)=|{tilde over (X)}_(T)(t,v)|². Itfollows from the above considerations that the spectral components ofthe signal power exhibit strong correlations whenever the spectralcomponents of the signal amplitude are strongly correlated.Consequently, modulated optical signals encoded with ASK, BPSK, ordinaryQPSK and 16-QAM formats exhibit strong correlations of the time-varyingoptical power components at frequencies f±α₀/2. An example of the strongcorrelations between the time-varying signal powers 71, 72 at f_(c)±α₀/2is shown in FIG. 7 for a noiseless 100 Gb/s PM-QPSK signal. It can beseen from FIG. 7 that the power variations 71, 72 at the two frequenciesoverlap completely, i.e. they are substantially identical.

It should be noted that signals encoded with ASK and BPSK modulationexhibit additional correlations of the spectral power components beyondthose found for the spectral amplitude components. It can be shown, forexample, that the spectral power components of ASK and BPSK signals arecorrelated at any two pairs of frequencies f_(c)±α/2 within thebandwidth of the signal, i.e. for arbitrary offset frequency α.

Just like for the optical amplitudes, the spectral components of theoptical power levels of random ASE noise do not exhibit any significantcorrelations. When optical noise is added to a modulated signal,therefore, the correlation between the spectral components of the noisysignal power decreases with decreasing OSNR. An example of the reducedspectral correlations between the optical powers at f_(c)±α₀/2 is shownin FIG. 8 for a noisy 100 Gb/s PM-QPSK signal having an OSNR of only 12dB. It can be seen from this graph that variations 81, 82 of the twooptical powers with time are substantially different.

Furthermore, it can be shown that the spectral correlations of theoptical signal power are much less sensitive to waveform distortionscaused by PMD and CD than the correlations of the optical signalamplitudes. In general, GVD from CD and/or DGD from PMD may introduce adifferential time delay Δt between the time-varying optical powermeasurements {tilde over (P)}(t,f−α/2) and {tilde over (P)}(t,f+α/2),leading to signals of the form {tilde over (P)}(t−Δt/2,f−α/2) and {tildeover (P)}(t+Δt/2,f+α/2), as shown in FIG. 9 for the example of a 100Gb/s PM-QPSK signal distorted by 50 ns/nm GVD. Two curves 91, 92 in FIG.9 are shifted in time by about 10 ns, but are otherwise substantiallyidentical. Accordingly, these differential time delays may substantiallyreduce the spectral correlations in a noise-free signal and, hence, mayneed to be compensated prior to calculating the correlation between thetwo signals. This may be accomplished, for example, by introducing avariable time-delay between the two signals and by adaptively varyingthis delay until the correlation between the two signals is maximal.

For the purpose of the present disclosure, it is advantageous to definea normalized power SCDF for the spectral power components {tilde over(P)}(t,f−α/2)=|{tilde over (X)}_(T)(t,f−α/2)|² and {tilde over(P)}(t,f+α/2)=|{tilde over (X)}_(T)(t,f+α/2)|² of the noisy signal,analogous to the normalized amplitude SCDF described above:

${{\hat{S}}_{p}^{\alpha}(f)} \equiv {\frac{\langle{{\overset{\sim}{P}\left( {t,{f + {\alpha/2}}} \right)} \cdot {\overset{\sim}{P}\left( {t,{f - {\alpha/2}}} \right)}}\rangle}{\sqrt{\langle\left\lbrack {\overset{\sim}{P}\left( {t,{f + {\alpha/2}}} \right)} \right\rbrack^{2}\rangle}\sqrt{\langle\left\lbrack {\overset{\sim}{P}\left( {t,{f + {\alpha/2}}} \right)} \right\rbrack^{2}\rangle}}.}$

It can be shown that noiseless signals encoded with ASK-, BPSK-,ordinary QPSK- and 16-QAM modulation exhibit Ŝ_(p) ^(α) ⁰ (f)=1 at anytwo frequency pairs f±α₀/2 which are within the optical bandwidth of thetransmitted signal. For modulated signals with added ASE noise, thenormalized power SCDF is always less than unity, i.e. Ŝ_(p) ^(α) ⁰(f)<1, just like the normalized amplitude SCDF. The normalized powerSCDF for noisy signals can be calculated analytically as describedbelow.

In the case of α>>1/T, which is of interest in the present disclosure,the numerator of Ŝ_(p) ^(α)(f) can be expanded into the following terms

{tilde over (P)}(t,f+α/2)·{tilde over (P)}(t,f−α/2)

=

P _(S)(t,f+α/2)·P _(S)(t,f−α/2)

+

P _(S)(t,f+α/2)·P _(N)(t,f−α/2)

+

P _(N)(t,f+α/2)·P _(S)(t,f−α/2)

+

P _(N)(t,f+α/2)·P _(N)(t,f−α/2)

,

wherein

P _(S)(t,f+α/2)·P _(N)(t,f−α/2)

=

P _(S)(t,f+α/2)

P _(N)(t,f−α/2)

,

and likewise

P _(N)(t,f+α/2)·P _(S)(t,f−α/2)

=

P _(N)(t,f+α/2)

·

P _(S)(t,f−α/2)

,

because the time-varying signal amplitudes are not correlated with therandom variations of the noise amplitudes. Furthermore, since the randomnoise power variations at substantially different optical frequenciesare uncorrelated, one has

P _(N)(t,f+α/2)·P _(N)(t,f−α/2)

=

P _(N)(t,f+α/2)

·

P _(N)(t,f−α/2)

=

P _(N)(t,f+α/2)

².

Similarly, the two terms in the denominator of Ŝ_(p) ^(α)(f) can beexpressed as

[{tilde over (P)}(t,f+α/2)]²

=

[P _(S)(t,f+α/2)]²

+

[P _(N)(t,f+α/2)]²

+3

P _(S)(t,f+α/2)

·

P _(N)(t,f+α/2)

[{tilde over (P)}(t,f−α/2)]²

=

[P _(S)(t,f−α/2)]²

+

[P _(N)(t,f−α/2)]²

+3

P _(S)(t,f−α/2)

·

P _(N)(t,f−α/2)

,

wherein the third term on the right side of these equations includes thecontribution

{Re[X _(T)(t,f±α/2)·N _(T)*(t,f±α/2)]}²

=¼

P _(S)(t,f+α/2)

·

P _(N)(t,f±α/2)

.

Therefore, in the special case of f=f_(c) and α=α₀, the power SCDF canbe written in the form

${{{\hat{S}}_{p}^{\alpha_{0}}\left( f_{c} \right)} = \frac{{\langle{P_{S}^{2}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} + {2{{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}\rangle}^{2}}{{\langle{P_{S}^{2}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} + {3{{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}^{2}\rangle}}},{wherein}$⟨P_(N)⟩ = ⟨P_(N)(t, f_(c) ± α₀/2)⟩ = ⟨P_(N)(t, f_(c) − α₀/2)⟩, ⟨P_(N)²⟩ = ⟨[P_(N)(t, f_(c) ± α₀/2)]²⟩ = ⟨[P_(N)(t, f_(c) − α₀/2)]²⟩

The above expression for the power SCDF contains two unknown andpotentially large quantities

P_(S) ²

and

P_(N) ²

, which cannot be directly measured or obtained from the measurements ofthe combined signal and noise powers at frequencies f+α/2 and f−α/2.

It is known that the averaged squared noise power

P_(N) ²

is related to the average noise power

P_(N)

as

P_(N) ²

=1.5

P_(N)

², because of the statistical properties of random ASE noise.) Withreference to J. W. Goodman, Statistical Optics (John Wiley and Sons, NewYork 1985), optical ASE noise can be described as a Gaussian randomprocess, which remains a Gaussian random process even after arbitrarylinear optical filtering (e.g. before or in the optical receiver). Thus,the relation

P_(N) ²

=1.5

P_(N)

² can be obtained from the known second and fourth moments of a Gaussianrandom process.

Furthermore, according to the present disclosure, the quantity

P_(S) ²

can be eliminated by multiplying the complementary power SCDF,

${{1 + {{\hat{S}}_{p}^{\alpha_{0}}\left( f_{c} \right)}} = \frac{{\langle P_{N}^{2}\rangle} - {\langle P_{N}\rangle}^{2} + {{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} \cdot {\langle P_{N}\rangle}}}{{\langle{P_{S}^{2}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} + {3{{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}^{2}\rangle}}},$

with the dimensionless factor D(f_(c),α₀), defined as

${{D\left( {f,\alpha} \right)} = \frac{\sqrt{\langle\left\lbrack {\overset{\sim}{P}\left( {t,{f + {\alpha/2}}} \right)} \right\rbrack^{2}\rangle}\sqrt{\langle\left\lbrack {\overset{\sim}{P}\left( {t,{f - {\alpha/2}}} \right)} \right\rbrack^{2}\rangle}}{{\langle{\overset{\sim}{P}\left( {t,{f + {\alpha/2}}} \right)}\rangle} \cdot {\langle{\overset{\sim}{P}\left( {t,{f - {\alpha/2}}} \right)}\rangle}}},$

which can be readily calculated from the measured signal and noisepowers {tilde over (P)}(t,f±α/2).

At optical frequencies f_(c)+α₀/2 and f_(c)−α₀/2, one obtains

${{D\left( {f_{c},\alpha_{0}} \right)} = \frac{{\langle{P_{S}^{2}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} + {3{{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}^{2}\rangle}}{{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle}^{2} + {2{{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}\rangle}^{2}}},$

and after multiplication with the complementary power SCDF

${{\hat{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)} = {{\left\lbrack {1 - {{\hat{S}}_{p}^{\alpha_{0}}\left( f_{c} \right)}} \right\rbrack {D\left( {f_{c},\alpha_{0}} \right)}} = {\frac{{0.5{\langle P_{N}\rangle}^{2}} + {{\langle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\rangle} \cdot {\langle P_{N}\rangle}}}{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}^{2}}.}}$

This equation can be solved for

P_(N)

${\frac{\langle P_{N}\rangle}{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle} = {1 - \sqrt{1 - {2{{\hat{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}2}}}},$

and substituted into the equation for OSNR to produce

$\begin{matrix}{{OSNR} = {{\frac{\sum\limits_{i}^{\;}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}} \cdot \frac{1}{1 - \sqrt{1 - {2{{\hat{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}}}}} - {\sum\limits_{i}^{\;}\frac{B_{meas}}{B_{noise}}}}} \\{{= \frac{\sum\limits_{i}^{\;}{\begin{Bmatrix}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle} -} \\{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}\left\lbrack {1 - \sqrt{1 - {2{{\hat{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}}}} \right\rbrack}\end{Bmatrix}B_{meas}}}{{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}\left\lbrack {1 - \sqrt{1 - {2{{\hat{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}}}} \right\rbrack}B_{noise}}},}\end{matrix}$

wherein again the summation extends over the entire spectral width ofthe transmitted optical signal.

Therefore, the in-band OSNR in a noisy signal can be determined from aspectral correlation measurement of the signal powers at opticalfrequencies f_(c)+α₀/2 and f_(c)−α₀/2. Just like in the case of usingcorrelation measurements of the signal's amplitude, these measurementsdo not require foreknowledge of the signal's modulation format, bitrate,or waveform.

For large OSNR values (e.g. larger than 15 dB for a 100 Gb/s PM RZ-QPSKsignal), one may approximate the in-band OSNR by the simpler expression

$\begin{matrix}{{OSNR} \approx {{\frac{\sum\limits_{i}^{\;}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}} \cdot \left\lbrack {\frac{1}{{\hat{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)} - 1} \right\rbrack} - {\sum\limits_{i}^{\;}\frac{B_{meas}}{B_{noise}}}}} \\{\approx {{\frac{\sum\limits_{i}^{\;}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}} \cdot \frac{1}{{\hat{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}} - {\sum\limits_{i}^{\;}\frac{B_{meas}}{B_{noise}}}}} \\{{= \frac{\sum\limits_{i}^{\;}{\left\{ {{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle} - {{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}{{\hat{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}}} \right\} B_{meas}}}{{\langle{{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}{{\hat{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}B_{noise}}},}\end{matrix}$

because

P_(N)

<<

P_(S)

, so that Ŝ_(p) ^(α) ⁰ (f_(c))≈1 and hence {circumflex over (D)}_(p)^(α) ⁰ (f_(c))<<1. FIG. 10 displays a numerical simulation of thereference in-band OSNR 100A determined from the above equation for a 100Gb/s PM-QPSK signal. The reference in-band OSNR 100A is shown with athick solid line. A normalized power SCDF 101 at f_(c)±α₀/2 is alsoshown. It is seen that an estimated in-band OSNR 100B, shown with adashed line, is substantially equal to the reference in-band OSNR 100Aover a range from about 15 dB to at least 30 dB.

Importantly, in-band OSNR may be determined from a spectral correlationmeasurement of transmitted signal powers at any frequency pair f±α₀/2within the spectral bandwidth of the signal, similar to the spectralcorrelation measurement of the transmitted signal amplitudes describedabove. Letting again

P_(S)(f+α₀/2)

=C

P_(S)(f−α₀/2)

≡C

P_(S)

, with C>1 being a real number, the normalized SCDF of the signal powershas the general form

${{{\hat{S}}_{p}^{\alpha_{0}}(f)} = \frac{{C{\langle P_{S}^{2}\rangle}} + {\left( {1 + C} \right){{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}\rangle}^{2}}{\sqrt{{\langle P_{S}^{2}\rangle} + {3{{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}^{2}\rangle}}\sqrt{{C^{2}{\langle P_{S}^{2}\rangle}} + {3C{{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}^{2}\rangle}}}},$

wherein

P_(N) ²

=1.5

P_(N)

². It is possible to eliminate the unknown quantity

P_(S) ²

from the above equation, using the dimensionless factor

${{D\left( {f,\alpha_{0}} \right)} = \frac{\sqrt{{\langle P_{S}^{2}\rangle} + {3{{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}^{2}\rangle}}\sqrt{{C^{2}{\langle P_{S}^{2}\rangle}} + {3C{{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}} + {\langle P_{N}^{2}\rangle}}}{{\langle{{CP}_{S} + P_{N}}\rangle}{\langle{P_{S} + P_{N}}\rangle}}},$

but the results are fairly complex.

For signals with relatively large OSNR (e.g. above 15 dB for a 100 Gb/sPM RZ-QPSK signal), one may neglect the terms proportional to

P_(N)

² and

P_(N) ²

, so that the complementary SCDF may be simplified to

${{1 - {{\hat{S}}_{p}^{\alpha_{0}}(f)}} \approx \frac{0.5\left( {1 + C} \right){{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}}{\sqrt{{\langle P_{S}^{2}\rangle} + {3{{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}}}\sqrt{{C^{2}{\langle P_{S}^{2}\rangle}} + {3C{{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}}}}},$

so that

${{\hat{D}}_{p}^{\alpha_{0}}(f)} \equiv {\left\lbrack {1 - {{\hat{S}}_{p}^{\alpha_{0}}(f)}} \right\rbrack {D\left( {f,\alpha_{0}} \right)}} \approx {\frac{0.5\left( {1 + C} \right){{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}}{{C{\langle P_{S}\rangle}^{2}} + {\left( {C + 1} \right){{\langle P_{S}\rangle} \cdot {\langle P_{N}\rangle}}}}.}$

This expression can be readily solved for

P_(N)

, yielding

${{\langle P_{N}\rangle} \approx {\frac{2C{{\hat{D}}_{p}^{\alpha_{0}}(f)}}{1 + C - {2{{\hat{D}}_{p}^{\alpha_{0}}(f)}}}{\langle{P_{S} + P_{N}}\rangle}}},$

and subsequently substituted into the equation for the OSNR:

$\begin{matrix}{{OSNR} \approx {{\frac{\sum\limits_{i}^{\;}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle{{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}} \cdot \left\{ {\frac{1 + C}{2C\; {{\hat{D}}_{p}^{\alpha_{0}}(f)}} - \frac{1}{C}} \right\}} - {\sum\limits_{i}^{\;}\frac{B_{meas}}{B_{noise}}}}} \\{\approx {{\frac{\sum\limits_{i}^{\;}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle}B_{meas}}}{{\langle{{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}} \cdot \frac{1 + C}{2C\; {{\hat{D}}_{p}^{\alpha_{0}}(f)}}} - {\sum\limits_{i}^{\;}\frac{B_{meas}}{B_{noise}}}}} \\{= {\frac{\sum\limits_{i}^{\;}{\begin{Bmatrix}{{\langle{{P_{S}\left( f_{i} \right)} + P_{N}}\rangle} -} \\{\frac{2C\; {{\hat{D}}_{p}^{\alpha_{0}}(f)}}{1 + C - {2{{\hat{D}}_{p}^{\alpha_{0}}(f)}}}{\langle{{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}}\rangle}}\end{Bmatrix}B_{meas}}}{\frac{2C\; {{\hat{D}}_{p}^{\alpha_{0}}(f)}}{1 + C}{\langle{{P_{S}\left( {f \pm {\alpha_{0}/2}} \right)} + P_{N}}\rangle}B_{noise}}.}}\end{matrix}$

FIG. 11 shows an example of OSNR curves 110A, 110B estimated from anumerically simulated noisy 100 Gb/s PM-QPSK signal having an OSNR of 15dB. The graph also displays a normalized power SCDF 111 as a function ofthe frequency offset f−f_(c). The OSNR 110A has been calculated from theabove equation, where C is calculated directly from the noisy signalspectrum, i.e. C≈

P_(S)(f−α₀/2)+P_(N)

/

P_(S)(f+α₀/2)+P_(N)

(the OSNR 110A is shown with a bold solid line). It can be seen fromthis graph that the above approximation slightly underestimates the OSNRat offset frequencies beyond 8 GHz, where the magnitude of the signalpower

P_(S)(f+α₀/2)) becomes comparable to that of the noise power

P_(N)

, as seen in FIG. 5, so that terms proportional to

P_(N)

² may not be neglected and the above approximation for the in-band OSNRis no longer valid. For comparison, the dashed curve 110B in FIG. 11displays the in-band OSNR calculated from the formula derived for C=1.

Amplitudes and phases of the spectral signal components, preferably atoptical frequencies f+α₀/2 and f−α₀/2 as described above, may bedetected simultaneously by two parallel coherent optical receivers, eachhaving phase and polarization diversity. With reference to FIG. 12, anexemplary apparatus 120 for determining OSNR of a modulated opticalsignal 121 may generally include a frequency selective splitter 122, ameasuring unit 123, and a digital signal processor (DSP) 124. Themodulated optical signal 121 contains a plurality of wavelengthchannels, e.g. wavelength division multiplexed (WDM) channels, or denseWDM (DWDM) channels, and is normally obtained at a test point along thecommunication link. An optional spectrum analyzer, not shown, may beused for measuring an optical power spectrum of the modulated opticalsignal 121. To be useful in the measurement, the optical power spectrumshould have at least one of the plurality of wavelength channels.

In the embodiment shown, the frequency selective splitter 122 is adual-channel coherent receiver having a tunable local oscillator lightsource, e.g. a pair of cw tunable lasers (“local oscillators”, or LOlasers) 125A and 125B tuned to first and second frequencies, for examplef+α₀/2 and f−α₀/2 respectively. The frequency selective splitter 122further includes a 3 dB splitter 119A and four polarizationbeamsplitters (PBS) 119B optically coupled as shown. The modulatedoptical signal 121 is mixed with the corresponding local oscillatorlaser signals in hybrid mixers 126. The measuring unit 123 includesdifferential photodetectors 127, low-pass filters (LPFs) 128, andanalog-to-digital converters (ADC) 129. Other suitable configurationsmay be used.

In operation, the frequency selective splitter 122 selects portions ofthe modulated optical signal 121 at first and second predeterminedoptical frequencies, in a selected one of the plurality of wavelengthchannels. The first and second predetermined optical frequencies areseparated by a non-zero frequency interval, e.g. α₀ as explained above.To that end, the modulated optical signal 121 is divided into twoidentical copies by means of the 3 dB power splitter 119A, which arethen coupled into the two substantially identical coherent receiver“channels”, i.e. receivers 122A and 122B. Each copy of the modulatedsignal 121 is mixed with a highly coherent light from one of the tunablelasers 125A and 125B outputting a highly coherent continuous-waveoptical signal at a predetermined optical frequency. The frequencies ofthe two local oscillator lasers are set to be substantially equal to thetwo frequencies at which the spectral correlation is to be measured,i.e. to f₁=f+α₀/2 and f₂=f−α₀/2, respectively. The tunable lasers 125Aand 125B preferably have a narrow linewidth, e.g. no greater than 100kHz.

Prior to mixing with the output light of the tunable lasers 125A and125B, each of the two copies of the modulated signal is decomposed intotwo signals having mutually orthogonal polarization states by means ofthe PBS 119B. Furthermore, each of the two orthogonally polarizedsignals is further split into two identical copies and thenindependently mixed with the output light from the tunable LO lasers125A and 125B, by means of the 90° hybrid mixers 126. The first copy ismixed with the LO light having an arbitrary optical phase and the secondcopy with the LO light having an optical phase shifted by 90° relativeto the LO light used for the first copy. Such a receiver is referred toherein as a coherent receiver having polarization and phase diversity.

The measuring unit 123 measures time-varying optical amplitudes andphases of the first and second portions of the modulated optical signal121. To that end, the four different mixing products in each of the twocoherent receiver channels 122A and 122B are then detected by balancedphoto-detectors 127, which convert the coherently mixed opticalamplitudes of the received signal and LO laser into a proportionalelectrical current, but substantially reject all non-coherently detectedsignal powers. The four time-varying detector currents i_(k)(t), k=1, .. . , 4, are subsequently amplified and filtered by the four identicalelectrical LPFs 128, before they are converted into digital signals by aset of four high-resolution ADCs 129, preferably having an effectivenumber of bits (ENOB) of at least 12 and a sampling rate substantiallyhigher than two times the bandwidth of LPFs.

Once the time-varying amplitudes and phases have been measured, thesignal processor 124 determines a correlation between the time-varyingamplitudes and phases of the first and second portions of the modulatedoptical signal, and calculates the OSNR from the correlation of thetime-varying parameters and the power spectrum of the modulated opticalsignal 121. This may be done as follows. The four receiver currentsi₁(t,f±α/2), . . . , i₄(t,f±α/2), generated in each of the two coherentreceiver channels 122A and 122B, describe the amplitude, phase, andpolarization state of the received optical signal at frequencies f±α/2.The four currents may be used to form a two-dimensional complex Jonesvector that is proportional to the Jones vector of the optical signal,i.e.

${{{\overset{\sim}{X}}_{T}\left( {t,{f \pm {\alpha/2}}} \right)} = {\beta \begin{bmatrix}{{i_{1}\left( {t,{f \pm {\alpha/2}}} \right)} + {{ji}_{2}\left( {t,{f \pm {\alpha/2}}} \right)}} \\{{i_{3}\left( {t,{f \pm {\alpha/2}}} \right)} + {{ji}_{4}\left( {t,{f \pm {\alpha/2}}} \right)}}\end{bmatrix}}},$

where β is an undetermined proportionality factor and j=√{square rootover (−1)}. Hence, the Jones vectors formed by the four receivercurrents in each of the two coherent receivers may be used to calculatethe normalized amplitude SCDF. Preferably, this is accomplished in afast digital signal processor (DSP), which processes the data at thesampling rate of the ADC 129.

Prior to calculating the amplitude SCDF, any undesired differentialphase shifts between the two Jones vectors {tilde over (X)}_(T)(t,f−α/2)and {tilde over (X)}_(T)(t,f+α/2) 124C may be removed (or compensated)by digital signal processing, as described above, including differentialphase shifts introduced by CD in the communication systems (“CDcompensation”) as well as static phase shifts introduced in thereceiver. This is done by a CD compensation module 124A. Likewise, anyPMD-induced differential polarization transformations between the twoJones vectors may be removed (“PMD compensation”). This is done by a PMDcompensation module 124B. This digital CD and PMD compensation may beaccomplished iteratively by means of a feedback loop, in which thedifferential phase shifts and the differential polarizationtransformations are varied in sufficiently small steps until theamplitude SCDF reaches a maximum. Then, the normalized amplitudecorrelation coefficient is computed by a computing module 124D.

For in-band OSNR measurements in QPSK and higher-order M-ary QAMsignals, the electrical bandwidth of the coherent receiver 122, which issubstantially equal to twice the bandwidth of the LPF 128, shallpreferably be as small as possible, so that the correlation between thetwo detected Jones vectors is maximal. On the other hand, the receiverbandwidth has to be wide enough so that a sufficiently large electricalsignal is available for determining the SCDF. Preferably, the 3-dBelectrical bandwidth of the LPF 128 should be around 100 MHz for in-bandOSNR measurements in 40 Gb/s and 100 Gb/s PM-QPSK signals. FIG. 13illustrates effects of the receiver bandwidth on the accuracy of OSNRmeasurements on a coherently detected 100 Gb/s PM RZ-QPSK signal for the3 dB bandwidth of the LPF 128 of 0.1 GHz, 0.4 GHz, 1 GHz, and 4 GHz.Severe degradations of the measurement accuracy are seen at high OSNRvalues when the receiver bandwidth substantially exceeds 400 MHz. Itshould be noted that the bandwidth requirement for the LPF 128 may scalelinearly with the bit-rate of the signal 121.

Furthermore, the optical frequencies of the two tunable (LO) lasers 125Aand 125B needs to be precisely adjusted in order to measure maximalcorrelation between the spectral components of the analyzed opticalsignal 121. This can be accomplished by first setting one of the two LOlasers (e.g. 125A) to a fixed frequency within the spectral bandwidth ofthe signal, and by then varying the optical frequency of the other LOlaser (e.g. 125B) continuously or in sufficiently small steps until theSCDF reaches a maximum. It is to be noted that a maximal amplitudecorrelation can only be observed over a very narrow optical frequencyrange, e.g. of less than 1 MHz. FIG. 14 displays a numerical simulationof an amplitude SCDF 141 as a function of the frequency separation ofthe two LO lasers 125A, 125B (FIG. 12) for the case of a 100 Gb/sPM-QPSK signal. In contrast, maximal power correlation curves 142A, 142B(FIG. 14) can be observed over a frequency range of several MHz, whichscales linearly with the electrical bandwidth of the receiver.Consequently, spectral correlation measurements require LO lasers withhigh frequency stability and low frequency noise. Typically, LO lasers125A, 125B having a linewidth of less than 10 kHz are preferable foramplitude correlation measurements, whereas LO lasers with a linewidthof less than 100 kHz may be sufficient for power correlationmeasurements. Solid curves 141, 142A correspond to an electricalreceiver with fifth-order Butterworth low-pass filter, and the dashedcurve 142B corresponds to a receiver with fifth-order Bessel low-passfilter, both having a 3-dB bandwidth of 40 MHz.

The embodiment of FIG. 12 may also be used to measure a power spectrumof the noisy signal (c.f. FIG. 3). This measurement is illustrated inFIG. 15. In an apparatus 150, only one of the two coherent receiverchannels 122A, 122B is needed e.g. 122A, including elements similar tothose described above with reference to FIG. 12. The desired powerspectrum may be obtained by scanning the LO laser 125A over the entirebandwidth of the signal 121, while recording the inner product of theJones vector, i.e. |X_(T)(t,f)|², as shown schematically in FIG. 15.Time-varying optical power levels of the first and second portions ofthe modulated optical signal 121 are used instead of the time-varyingoptical amplitudes and phases.

FIG. 16 displays a schematic diagram of an apparatus 160 for the abovedescribed in-band OSNR measurements using the power SCDF of the opticalsignal. Similarly to the apparatus 120 of FIG. 12, the apparatus 160 ofFIG. 16 has two parallel coherent receiver channels 122A, 122B, whichare connected to a fast DSP 164. One difference is that the DSP in FIG.16 uses modules 164C, 164D to calculate the correlation between thetime-varying powers of the two Jones vectors 164A, i.e.

$\begin{matrix}{{\overset{\sim}{P}\left( {t,{f \pm {\alpha/2}}} \right)} = {{{\overset{\sim}{X}}_{T}\left( {t,{f \pm {\alpha/2}}} \right)}}^{2}} \\{= {\beta^{2}\left\lbrack {{{i_{1}\left( {t,{f \pm {\alpha/2}}} \right)}}^{2} + {{i_{2}\left( {t,{f \pm {\alpha/2}}} \right)}}^{2} +} \right.}} \\{\left. {{{i_{3}\left( {t,{f \pm {\alpha/2}}} \right)}}^{2} + {{i_{4}\left( {t,{f \pm {\alpha/2}}} \right)}}^{2}} \right\rbrack.}\end{matrix}$

The requirements for the electrical bandwidth of the receiver areidentical to those described above for measurements of the amplitudeSCDF (c.f. FIG. 13).

Prior to calculating the power SCDF, any undesired differential timedelays between the two signal powers {tilde over (P)}(t,f−α/2) and{tilde over (P)}(t,f+α/2) may need to be removed by digital signalprocessing, as described above, including delays introduced by CD andPMD as well as those introduced in the receiver. This differential timedelay compensation may be accomplished iteratively by means of afeedback loop, in which the delay is varied in sufficiently small stepsuntil the power SCDF reaches a maximum. The differential time delaycompensation is performed by a module 164B.

Turning now to FIG. 17, an apparatus 170 may be used for measuringspectral correlations in power level or amplitude of the signal 121.Just like the embodiments of apparatus 120 of FIG. 12 and 160 of FIG.16, the apparatus 170 of FIG. 17 includes a frequency selective splitter172 including two parallel coherent receivers, or receiver channels172A, 172B; and a measuring unit 173 to measure correlations between twodistinct frequency components of the optical amplitude or power of thesignal 121. Unlike the homodyne receivers used in the apparatus 120 ofFIG. 12 and 160 of FIG. 16, the apparatus 170 of FIG. 17 uses twoheterodyne receivers, in which the frequencies of the two LO lasers 125Aand 125B are offset from f−α/2 and f+α/2 by an equal amount f_(IF),which is substantially larger than the bandwidth of the signals fromwhich the amplitude or power SCDF is calculated. As a result, thespectral components to be analyzed are found at intermediate frequencyf_(IF) in the receiver photo currents generated by the coherent mixerand, hence, can be selected by means of electrical band-pass filters(BPF) 178 having a pass band centered around f_(IF). Advantageously,such heterodyne receivers do not require coherent mixers with phasediversity, i.e. they do not require 90° hybrid mixers, but instead mayuse simple 3 dB optical couplers 179A to superimpose the optical signalwith the output light of the LO lasers 125A, 125B, yielding only twophoto currents in each of the two coherent receivers 172A, 172B.

The four signals needed to form the Jones vectors {tilde over(X)}_(T)(t,f±α/2) in each of the two coherent receivers 172A, 172B canbe recovered by down-converting the two bandpass-filtered photocurrentsby means of an electrical homodyne receiver to the baseband.Advantageously, this down-conversion and subsequent electrical filteringmay be accomplished by digital signal processing in a DSP 174, as shownschematically in FIG. 17, using digital signal mixers 175 and digitallow-pass filters 176. However, such digital down-conversion requiresthat the sampling rate of the ADCs 129 is substantially larger than 2f_(IF).

In another embodiment, the two independent LO lasers 125A, 125B for thetwo receivers 172A, 172B may be replaced by a single laser and anoptical frequency shifter. Referring now to FIG. 18, an apparatus 180 issimilar to the apparatus 160 of FIG. 16. In the apparatus 180 of FIG.18, output light 181 of the LO laser 125A is equally split into twosignals 181A and 181B, of which the first signal 181A serves as the LOsignal for the first coherent receiver channel 122A, whereas the secondsignal 181B is shifted in frequency by an amount substantially equal toa by means of an optical frequency shifter 183, to generate a LO opticalsignal for the second coherent receiver channel 122B.

The advantage of using the single LO laser 125A for both coherent mixerpairs 126 is that it substantially reduces the frequency stabilityrequirements for the LO laser 125A, provided that optical path lengthsfor the modulated signal between the 3 dB splitter 119A and the fourinputs to the 90° hybrid mixers 126 are substantially equal and,likewise, that the optical path lengths for the two LO signals betweenthe 3 dB splitter 179B and the four inputs to the 90° hybrid mixers 126are substantially equal. Another advantage of this embodiment is thatthe frequency difference between the two LO signals is determined by thefrequency offset generated in the optical frequency shifter 183, whichmay be set with much higher accuracy than the difference frequency oftwo independent LO lasers 125A and 125B in FIGS. 12 and 16.

Referring to FIG. 19A, an electro-optic frequency shifter 190A has beendescribed in detail by M. Izutsu et al. in “Integrated optical SSBmodulator/frequency shifter,” J. Quantum Electron. Vol. 17, p. 2225(1981), incorporated herein by reference, and will only be brieflydescribed herein. The electro-optic frequency shifter 190A includes anoptical modulator 193A coupled to a tunable laser source 194. Thefrequency offset, or the modulation frequency a is determined by theelectrical frequency of an sinusoidal oscillator 191 outputting twosignals of identical frequency that are phase-shifted by 90°. Theoptical output signal of the frequency shifter 190A is filtered by anoptical band-pass filter 192, as shown in FIG. 19A, to filter out otherundesired frequency components, i.e. harmonics of the modulationfrequency a.

Turning to FIGS. 19B and 19C, electro-optic frequency shifters 190B and190C both generate two shifted optical frequency components from asingle optical input signal provided by the tunable laser 194. Thefrequency shifter 190B of FIG. 19B includes a Mach-Zehnder modulator193B driven by the sinusoidal oscillator 191 with a sinusoidal signal offrequency α/2, and operated in such a way that it generates acarrier-suppressed optical line spectrum, as disclosed by F. Heismann etal. in U.S. Pat. No. 8,135,275 issued Mar. 13, 2012 “Measuring chromaticdispersion in an optical wavelength channel of an optical fiber link”,incorporated herein by reference. The two main frequency components inthe output of the Mach-Zehnder modulator at frequencies f−α/2 and f+α/2may be separated from each other and the other undesired frequencycomponents by an optical diplexer 195.

The frequency shifter 190C shown in FIG. 19C has two identical opticalmodulators 193A of the kind shown in FIG. 19A and, in general, generatesfewer undesired frequency components that the frequency shifter shown inFIG. 19B. However, it has substantially higher optical insertion lossthan the simpler frequency shifter 190B of FIG. 19B.

Turning now to FIG. 20, another embodiment of an apparatus 200 formeasuring correlations in the signal's optical power spectrum is shown.The apparatus 200 includes two narrowband optical band-pass filters, 201(OBPF1) and 202 (OBPF2), for selecting spectral components to beanalyzed, two photodetectors 127, two LPFs 128, and two ADCs 129. Inoperation, optical power levels of the two spectral components filteredby the OBPF1 201 and OBPF2 202, are detected by the photodetectors 127,filtered by the LPFs 128, and digitized by the ADCs 129. A processingunit 204 performs differential group delay compensation (module 204A)and then calculates a normalized power correlation coefficient (module204B).

The apparatus 200 is substantially simpler than the apparatus 160 ofFIG. 16, but the apparatus 200 requires OBPFs with a 3-dB opticalbandwidth of less than 1 GHz. Furthermore, the OBPF1 201 and OBPF2 202shall not exhibit significant polarization dependence, such aspolarization-dependent loss or polarization-dependent shifts of theoptical passband.

Exemplary methods for determining OSNR and other signal degradationparameters of a modulated optical signal, e.g. the WDM/DWDM signal 121of FIG. 12, will now be briefly described. Referring to FIG. 21, amethod 210 for determining an OSNR of a modulated optical signalpropagating in a transmission link is presented. In a step 211, anoptical power spectrum of the modulated optical signal is measured. Theoptical power spectrum includes at least one of the plurality ofwavelength channels. In a following step 212, a time-varying parameteris measured during a measurement time. As explained above, thetime-varying parameter may include at least one of: time-varying opticalsignal amplitudes and phases in two mutually orthogonal polarizationstates; and time-varying optical signal intensities in two mutuallyorthogonal polarization states. The time-varying parameter is measuredsimultaneously at first and second predetermined optical frequencies ina selected one of the plurality of wavelength channels. The first andsecond predetermined optical frequencies are separated by a non-zerofrequency interval.

In a following step 213, a correlation between the time-varyingparameters measured in the previous step 212 at the first and secondoptical frequencies is determined. The correlation is determined bycalculating a correlation coefficient between the time-varyingparameters at the first and second optical frequencies. The correlationcoefficient is preferably normalized. In a following step 214, the OSNRis determined based on the optical power spectrum measured in the firststep 211 and the correlation coefficient calculated in the third step213. The measurement time of the measuring step 212 is preferablysufficiently long to reach a pre-determined level of fidelity (e.g. 90%)of the obtained correlation coefficient.

The correlation determining step 213 may include at least one of:removing differential phase and time delays introduced by chromaticdispersion in the transmission link between the time-varying parametersat the first and second optical frequencies; and removing a differentialgroup delay introduced by polarization mode dispersion in thetransmission link between the time-varying parameters at the first andsecond optical frequencies, as explained above.

Turning to FIG. 22, a method 220 for determining a group velocitydispersion accumulated due to chromatic dispersion of a modulatedoptical signal, e.g. the WDM/DWDM signal 121, is presented. The method220 includes a step 221 of measuring, during a measurement time,time-varying amplitudes and phases of the modulated optical signal intwo mutually orthogonal polarization states simultaneously at first andsecond predetermined optical frequencies separated by a non-zerofrequency interval, in at least one of the plurality of wavelengthchannels. In a next step 222, a differential time and phase delay isintroduced between the signals representing time-varying optical signalamplitudes and phases at the first and second optical frequencies. In anext step 223, a correlation between the time-varying optical signalamplitudes and phases at the predetermined optical frequencies isdetermined, by calculating a correlation coefficient between thetime-varying amplitudes and phases of the modulated optical signal. In anext step 224, the differential time and phase delay of the second step222 is varied.

The two last steps 223 and 224 are repeated until the correlationcoefficient reaches a maximum, and in a step 225, the group velocitydispersion is calculated from the differential time and phase delayintroduced in the second step 222 and varied in the fourth step 224, andthe frequency interval of the first step 221. Similarly to the previousmethod 210, the measurement time of the measuring step 221 is preferablysufficiently long to reach a pre-determined level of fidelity (e.g. 90%)of the calculated correlation coefficient.

Referring now to FIG. 23, a method 230 for determining a differentialgroup delay accumulated due to polarization mode dispersion of amodulated optical signal, e.g. the WDM/DWDM signal 121, is presented.The method 230 includes a step 231 of measuring, during a measurementtime, time-varying optical signal power levels of the modulated opticalsignal in two mutually orthogonal polarization states simultaneously atfirst and second predetermined optical frequencies separated by anon-zero frequency interval, in at least one of the plurality ofwavelength channels. In a next step 232, a differential group delay isintroduced between the signals representing the time-varying opticalsignal powers at the first and second optical frequencies. In a nextstep 233, a correlation is determined between the time-varying opticalsignal powers at the first and second optical frequencies, bycalculating a correlation coefficient between the time-varying opticalsignal powers. In a next step 234, the differential group delay of thesecond step 232 is varied. The two last steps 233 and 234 are repeateduntil the correlation coefficient reaches a maximum. The last iteratedvalue of the differential group delay is used to obtain the differentialgroup delay accumulated due to polarization mode dispersion of themodulated optical signal. Similarly to the previous method 220, themeasurement time of the measuring step 231 is preferably sufficientlylong to reach a pre-determined level of fidelity (e.g. 90%) of thecomputed correlation coefficient.

The hardware used to implement the various illustrative logics, logicalblocks, modules, and circuits described in connection with the aspectsdisclosed herein may be implemented or performed with a general purposeprocessor, a digital signal processor (DSP), an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA) orother programmable logic device, discrete gate or transistor logic,discrete hardware components, or any combination thereof designed toperform the functions described herein. A general-purpose processor maybe a microprocessor, but, in the alternative, the processor may be anyconventional processor, controller, microcontroller, or state machine. Aprocessor may also be implemented as a combination of computing devices,e.g., a combination of a DSP and a microprocessor, a plurality ofmicroprocessors, one or more microprocessors in conjunction with a DSPcore, or any other such configuration. Alternatively, some steps ormethods may be performed by circuitry that is specific to a givenfunction.

The present disclosure is not to be limited in scope by the specificembodiments described herein. Indeed, other various embodiments andmodifications, in addition to those described herein, will be apparentto those of ordinary skill in the art from the foregoing description andaccompanying drawings. Thus, such other embodiments and modificationsare intended to fall within the scope of the present disclosure.Further, although the present disclosure has been described herein inthe context of a particular implementation in a particular environmentfor a particular purpose, those of ordinary skill in the art willrecognize that its usefulness is not limited thereto and that thepresent disclosure may be beneficially implemented in any number ofenvironments for any number of purposes. Accordingly, the claims setforth below should be construed in view of the full breadth and spiritof the present disclosure as described herein.

1-5. (canceled)
 6. A method for determining a group velocity dispersionaccumulated due to chromatic dispersion of a modulated optical signalcomprising a plurality of wavelength channels, the method comprising:measuring time-varying amplitudes and phases of the modulated opticalsignal in two mutually orthogonal polarization states simultaneously atfirst and second predetermined optical frequencies separated by anon-zero frequency interval, in at least one of the plurality ofwavelength channels; introducing a differential time and phase delaybetween signals representing the time-varying optical signal amplitudesand phases at the first and second optical frequencies; determining acorrelation between the time-varying optical signal amplitudes andphases at the predetermined optical frequencies by calculating acorrelation coefficient between the time-varying amplitudes and phasesof the modulated optical signal; varying the differential time and phasedelay; determining the correlation between the time-varying opticalsignal amplitudes and phases at the predetermined optical frequenciesand varying the differential time and phase delay until the correlationcoefficient reaches a maximum; and calculating the group velocitydispersion from the differential time and phase delay, and the frequencyinterval.
 7. A method for determining a differential group delayaccumulated due to polarization mode dispersion of a modulated opticalsignal comprising a plurality of wavelength channels, the methodcomprising: measuring time-varying optical signal power of the modulatedoptical signal in two mutually orthogonal polarization statessimultaneously at first and second predetermined optical frequenciesseparated by a non-zero frequency interval, in at least one of theplurality of wavelength channels; introducing a differential group delaybetween signals representing the time-varying optical signal powers atthe first and second optical frequencies; determining a correlationbetween the time-varying optical signal powers at the first and secondoptical frequencies by calculating a correlation coefficient between thetime-varying optical signal powers; varying the differential groupdelay; determining a correlation between the time-varying optical signalpowers at the first and second optical frequencies and varying thedifferential group delay until the correlation coefficient reaches amaximum; and using the last value of the differential group delay variedto obtain the differential group delay accumulated due to polarizationmode dispersion of the modulated optical signal. 8-23. (canceled) 24.The method according to claim 6, wherein the frequency interval issubstantially equal to a symbol repetition frequency of the modulatedoptical signal in the selected wavelength channel, or an integermultiple thereof.
 25. The method according to claim 6, whereindifferences between each of the first and second optical frequencies anda carrier frequency of the modulated optical signal in the selectedwavelength channel are substantially of equal magnitude.
 26. The methodaccording to claim 6, determining a correlation between the time-varyingparameters measured for the first and second optical frequencies basedon at least one of: removing differential phase and time delaysintroduced by chromatic dispersion in the transmission link between thetime-varying parameters at the first and second optical frequencies; andremoving the differential group delay.
 27. The method according to claim6, wherein the time-varying parameter comprises the time-varying opticalsignal amplitudes and phases.
 28. The method according to claim 7,wherein the frequency interval is substantially equal to a symbolrepetition frequency of the modulated optical signal in the selectedwavelength channel, or an integer multiple thereof.
 29. The methodaccording to claim 7, wherein differences between each of the first andsecond optical frequencies and a carrier frequency of the modulatedoptical signal in the selected wavelength channel are substantially ofequal magnitude.
 30. The method according to claim 7, determining acorrelation between the time-varying parameters measured for the firstand second optical frequencies based on at least one of: removingdifferential phase and time delays introduced by chromatic dispersion inthe transmission link between the time-varying parameters at the firstand second optical frequencies; and removing the differential groupdelay.
 31. The method according to claim 7, wherein the time-varyingparameter comprises the time-varying optical signal amplitudes andphases.
 32. An apparatus for determining an optical signal-to-noiseratio of a modulated optical signal comprising a plurality of wavelengthchannels, the apparatus comprising: a spectrum analyzer to measure anoptical power spectrum of the modulated optical signal; a frequencyselective splitter to select first and second portions of the modulatedoptical signal at first and second predetermined optical frequencies,wherein the first and second predetermined optical frequencies areseparated by a non-zero frequency interval; a measuring unit to measurea time-varying parameter comprising at least one of: time-varyingoptical amplitudes and phases; and time-varying optical power levels ofthe first and second portions of the modulated optical signal; and asignal processor to: determine a correlation between the time-varyingparameters of the first and second portions of the modulated opticalsignal; and calculate the optical signal-to-noise-ratio from thecorrelation of the time-varying parameters and the power spectrum of themodulated optical signal.
 33. The apparatus according to claim 32,wherein: the time-varying parameter comprises the time-varying opticalsignal amplitudes and phases; the amplitude and phase detector comprisesa coherent receiver, wherein the coherent receiver has phase andpolarization diversity; and the frequency selective splitter comprises atunable local oscillator light source, wherein the tunable localoscillator light source comprises a laser operating at a predeterminedoptical frequency, and a tunable optical frequency shifter opticallycoupled to the laser, for shifting the optical frequency of the laser.34. The apparatus according to claim 33, further comprising: an opticalpower splitter to split output light of the laser into first and secondportions, wherein the first portion of the output light is coupled tothe tunable frequency shifter, whose output signal is coupled to thecoherent receiver for detecting an amplitude and phase of the firstportion of the modulated optical signal, whereas the second portion ofthe output light is coupled directly to the coherent receiver fordetecting an amplitude and phase of the second portion of the modulatedoptical signal.
 35. The apparatus according to claim 34, wherein thetime-varying parameter comprises the time-varying optical power levelsof the first and second portions of the modulated optical signal, andwherein the measuring unit comprises a photo receiver for measuring thetime-varying optical power levels.
 36. The apparatus according to claim35, wherein the frequency selective splitter comprises a coherentreceiver comprising a tunable local oscillator light source, wherein thecoherent receiver has phase and polarization diversity.
 37. Theapparatus according to claim 36, wherein the tunable local oscillatorlight source comprises a laser operating at a predetermined opticalfrequency, and a tunable optical frequency shifter optically coupled tothe laser.